Introduction To Functions

Inverse Function (1.5)

Functions are represented with function notation "f(x)". Whereas, Inverse functions is the "inverse" of "f(x)", the Inverse function notation looks like "f-1(x)"

How to Find the Inverse Function:

We can find the Inverse algebraically, using x - y notation, you switch all the x's and y's within the equation.

For example if the equation was:

f(x) = 3x + 4

We would first put it into x - y notation

y = 3x + 4

Then we swap the x's and y's

x = 3y + 4

Now we simply just solve for "y"

y= x-4/3

Finally we just put it into the "Inverse Notation"

f-1(x) = x-4/3

Also if we were given a group of coordinate pairs {(1,3), (2,5), (3,7)}

We can just switch the x and y values so that the inverse of this pair would be:

{(3,1), (5,2), (7,3)}

Fun Fact!

The domain that belonged to "f(x)" becomes the range of "f-1(x)" and the range of "f(x)" becomes the domain of "f-1(x)".

So essentially if you swap the Domain and Range for "f(x)" you will get that functions inverse.

Domain & Range (1.4)

Function Notation (1.2)

Function Notation looks like: f(x).

f(x) essentially means "y".

To elaborate this, for Linear, the equation is:

y = x (x - y notation)

This has the same meaning as:

f(x) = x (function notation)

Essentially the f(x) just replaces y when it comes to functions because "f(x)" represents "y" and the "(x)" represents all the x values in the equations.

Lets say we needed to sub in a value to find "y", if we were given f(3) = .....
We would sub in the 3 into every x value within the equation and just simply use our math skills to solve the question.

Another way to look at this is that the "(x)" is the input, and "f(x)" would be the value of output.

Relations and Functions (1.1)

Relations

Relations can be:

A set of ordered pairs

{(1,3), (2,4), (1,7), (5,9)}

Functions

Functions can be:

A relation or set of orders pairs is where there can't be two same x values in a set of ordered pairs. Every x value has only one y value. Every function is a relation, but not every relation is a function.

How to Find if a relation is a function

If we have a set of ordered pairs:

{(1,3), (2,9), (1,4), (3,2)}

We can see that when x = 1, y is 3 and 4. Thus this can't be a function.

We can use a trick known as the "Vertical Line Test"

Parent Functions (1.3)

Parent Function:
f(x)=x

Characteristics:

Diagonal line that goes to positive and negative infinity.

Domain and Range have no restrictions due to the fact that the line is all real numbers.

Will have only one x and y intercept.

There is no max or min value since the goes on for infinity in both the positive and negative directions.

Passes VLT.

Transformations of Linear Function:

This function can take the form of:

y = mx + b

OR

f(x) = mx + b

This is called slope-intercept form with "b" being the y intercept and "m" being the rise over run (slope).

This graph can also have a reflection causing it to essentially flip vertically.

One fun fact is that no matter how we change our "b" or "m", the line will at most only maintain a presence in 3/4 quadrants.

Parent Function:
f(x)=x2

Characteristics:

U-shaped curved line either going to positive or negative infinity.

Domain will remain as all real numbers, but the Range will have a restriction depending on if the parabola is a max or min value. This means that no y values can go below the vertex if it is a min value and vice verse if it was a max. To clarify, Domain will not gain a restriction, Range will gain a restriction depending on if vertex is max or min.

Only case when Domain may have a restriction is when you deal with a word problem, that problem may be something like if a ball were thrown, let y-axis represent height and x-axis represent time, after 6s the ball hits the ground. Since (6,0) would be time of the ball reaching the ground, its the end of the story, the ball can't keep going below into the negative y values (-1m,-2m, etc...). This is why Domain would have a restriction in this case.

An axis of symmetry cuts the parabola in half by the vertex giving an identical shape on either side of the line. This line will intersect at only one point of the parabola which is the x coordinate of the vertex.

Passes VLT.

Transformations of Quadratic Function:

This Function can take the form of:

f(x) = ax2 + bx + c (Standard Form)

f(x) = a(x - h)2 + k (Vertex Form)

f(x) = a(x - r1)(x - r2) (Factored Form)

Standard form is usually the harder equation to gain information from regarding the Vertex or Roots. In order to easily/effectivly find the parabola on a graph given the standard form equation. You will either complete the square to convert to Vertex Form, or use Quadratic Formula to convert to Factored Form which will give you the Zeros/Roots/X-Intercepts of the parabola.

Parent Function:
f(x)=1/x

Characteristics:

This graph has two portions that never meet and never cross the y and x axis. This is also called a Vertical and Horizontal Asymptote.

Depending on the value inputted into the equation, it can have a vertical translation an horizontal translation which will create a new Vertical or Horizontal Asymptote depending on the values for translation.

Asymptote(s) are the line that the function will never cross.

Passes VLT.

Transformations of Reciprocal Function:

The graph can be transformed using the transformation equation, a vertical translation would give the function a new horizontal asymptote and a horizontal translation would give the graph a new vertical asymptote.

By adjusting our "a" for the graph we can see the curve when the line switches direction become more steep moving farther away from the origin of the cartesian plane or it can get super close.

Parent Function:
f(x)=/x

*Square Root doesn't show but using "/" as a replacement for the Square Root symbol*

Characteristics:

Looks like a Quadratic Function/Parabola flipped on its side and cut in half.

Parent Function starts at the origin (0,0) and continues in quadrant 1 to positive infinity.

Domain and Range would contain a restriction, Domain would say that there can be no values that are less then 0 on the x axis. So -1, -2, -3, etc...

The Range would say that there can be no values less then 0 on the y axis, so -1, -2. -3, etc... are not possible.

Passes VLT.

Transformations of Square Root Function:

Transformations on the Square Root Function may look like it being pulled upwards or squished closer to the x-axis. It can also flip/reflect through the y-axis and x-axis. These different factors will adjust the Domain and Range of these functions.

Parent Function:
f(x)=|x|

Characteristics:

The function continues in positive infinity when looking at the parent function (not including transformations like reflection).

Has a distinct V-shape with no "curve", straight lines like the linear function.

If you look at where the line goes on the graph, you can see that it increases by 1 as it grows. (+-1,1), (+-2,2), (+-3,3), etc...

Domain would not have a restriction since it keeps growing an infinite amount of times.

Range would have a restriction since depending on if the "a" is +-, it will have a max or min value just like the Quadratic Function. Restriction for the Range could say that there are no values y values that can be less than 0 since in this case, the vertex is a min.

Passes VLT.

Transformations of Absolute Value Function

Transformation can cause this function to experience a stretch and compression horizontally and vertically.

It can also be reflected in both the x and y axis(s).

The graph can be translated/shifted left and right, up and down. Depending on the "d" and "c" values.

Transformations of Parent Functions (1.6) (1.7)

Transformation Equation:

Each parent function can have transformations applied to it using the equation:

f(x) = +-a f[ +-k (x - d) ] + c

The "+-" in front of "a" represents a reflection in the x-axis

The "a" value represents if there will be a vertical stretch or compression

The "+-" in front of "k" represents a reflection in the y-axis

The "k" value represents if there will be a horizontal stretch or compression.

The "d" value represents horizontal translation/shift left or right. If the "d" value is negative, that means that the function will translate right, and if it's a positive, that means it will translate left. To help explain this, think of (x - 3), we can set this to be equal to zero so (x - 3) = 0. We want to isolate for x. So simply just move the 3 so the equation becomes x = 3. This is why you move the opposite direction when dealing with horizontal translations.

The "c" value represents vertical translation, so if it's a positive, you will shift up and if it's negative you will go down.

Depending on the values inputted into this equation, we can use this to adjust our mapping rule for whatever function we're dealing with. For example the quadratic function has key points such as (0,0) , (1,1) , (-1,1) , (2,4) , (-2,4) , (3,9) , (-3,9).

We can determine a mapping rule using the transformations. The general mapping rule when applying transformations is:

(x/k + d, ay +c)

With this equation, we would sub in whatever our key points were to this equation to solve for the new coordinate pair once transformations are applied.